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Magazine Chapter 2: Problem 475
Solve each inequality. Graph the solution and write the solution in interval notation. $$ |2 x-1|>5 $$
Short Answer
Expert verified
The solution is \((-\infty, -2) \cup (3, \infty)\).
Step by step solution
01
- Understand the absolute value inequality
The given inequality is \(|2x - 1| > 5\). This inequality can be split into two separate linear inequalities because an absolute value expression represents distance from zero.
02
- Split into two inequalities
The inequality \(|2x - 1| > 5\) can be written as two separate inequalities: 1) \(2x - 1 > 5\) 2) \(2x - 1 < -5\)
03
- Solve the first inequality
For \(2x - 1 > 5\): 1) Add 1 to both sides: \(2x > 6\) 2) Divide by 2: \(x > 3\)
04
- Solve the second inequality
For \(2x - 1 < -5\): 1) Add 1 to both sides: \(2x < -4\) 2) Divide by 2: \(x < -2\)
05
- Combine the inequalities
The solution to the inequality \(|2x - 1| > 5\) is the combination of \(x > 3\) and \(x < -2\).
06
- Write the solution in interval notation
The solution in interval notation is: \((-\infty, -2) \cup (3, \infty)\)
07
- Graph the solution
To graph the solution, draw a number line with an open circle on -2 and another open circle on 3. Shade the regions to the left of -2 and to the right of 3.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
absolute value
The concept of absolute value refers to the distance of a number from zero on a number line. It is always non-negative. The absolute value of a number \(x\) is written as \(|x|\). For example, \(|-3| = 3\) and \(|3| = 3\). In the context of inequalities, absolute value inequalities indicate a range of distances from a point. For instance, the inequality \(|2x - 1| > 5\) means that the expression \(2x - 1\) is more than 5 units away from 0.
linear inequalities
Linear inequalities are like linear equations, but instead of an equal sign, they use inequality symbols such as >, <, \(\geq\), and \(\leq\). To solve a linear inequality, follow the same steps as solving a linear equation: isolate the variable on one side.
Let's solve the inequalities from the original problem:
For \(2x - 1 > 5\):
Let's solve the inequalities from the original problem:
For \(2x - 1 > 5\):
- First, add 1 to both sides: \(2x > 6\)
- Then divide by 2: \(x > 3\)
- First, add 1 to both sides: \(2x < -4\)
- Then divide by 2: \(x < -2\)
interval notation
Interval notation is a way of expressing a set of numbers as an interval. It uses brackets and parentheses to show the range of values. For example:
- \([a, b]\) includes both endpoints a and b.
- \((a, b)\) does not include the endpoints.
- \([a, b)\) includes a but not b.
- \((a, b]\) includes b but not a.
- \(x < -2\) is expressed as \((-\infty, -2)\)
- \(x > 3\) is expressed as \((3, \infty)\)
graphing inequalities
Graphing inequalities helps to visualize the solution set on a number line or coordinate plane. We use open or closed circles and shaded regions to represent different parts of the solution.
In the given exercise, the solution \((-\infty, -2) \cup (3, \infty)\) can be graphed as follows:
In the given exercise, the solution \((-\infty, -2) \cup (3, \infty)\) can be graphed as follows:
- First, draw a number line.
- Place an open circle at -2 and another open circle at 3 to indicate these points are not included in the solution.
- Shade the region to the left of -2, indicating that all values less than -2 are part of the solution.
- Shade the region to the right of 3, indicating that all values greater than 3 are part of the solution.
